This paper is concerned with designing efficient algorithms for recovering sparse signals from noisy underdetermined measurements. More precisely, we consider minimization of a nonsmooth and nonconvex sparsity promoting function subject to an error constraint. To solve this problem, we use an alternating minimization penalty method, which ends up with an iterative proximal-projection approach. Furthermore, inspired by accelerated gradient schemes for solving convex problems, we equip the obtained algorithm with a so-called extrapolation step to boost its performance. Additionally, we prove its convergence to a critical point. Our extensive simulations on synthetic as well as real data verify that the proposed algorithm considerably outperforms some well-known and recently proposed algorithms.